Finite Group Databasebeta

← #10695426

Subgroups — #10695426

Order 768.

856 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 856 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4993C211yesmaximal, characteristic, complemented
2C2×#4968C211yesmaximal, characteristic, complemented
3#74031C211yesmaximal, characteristic
4C3×#1824C211yesmaximal, characteristic, complemented
5C2×#4470C211yesmaximal, characteristic, complemented
6C2×#4468C211yesmaximal, characteristic, complemented
7#67624C211yesmaximal, characteristic, complemented
8#9920—31nomaximal, Sylow
9C22×C3×C4⋊C4C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C22×(C2×D3)⋊C4—21no—
12C22×(C2×D3)⋊C4C2211yescharacteristic
13C22×Dic3⋊C4—21no—
14C2×#882—21no—
15C2×C3×C22⋊C8C2211yescharacteristic
16C3×#380C2211yescharacteristic
17C2×C3⋊#95C2211yescharacteristic
18C2×#830C2211yescharacteristic
19#4993—22no—
20#4993—22no—
21#4991—22no—
22#4968—22no—
23#4968—22no—
24C2×#809—21no—
25C2×#808—21no—
26#4958—21no—
27#4674—21no—
28#4671—21no—
29C3×#340—22no—
30#4470—22no—
31#4468—22no—
32C2×#481—31no—
33#2617—31no—
34C2×#399—31no—
35#1824D311yescharacteristic, complemented
36C2×#327—31no—
37C2×#323—31no—
38#1294—31no—
39C24×D3D411yescharacteristic
40C23×C3×C4—21no—
41C23×C3×C4C2311yescharacteristic
42C22×#67—21no—
43C23×Dic3—21no—
44C23×Dic3D411yescharacteristic
45C22×D12—22no—
46C22×D12D411yescharacteristic
47C22×C3×C8—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4D412yes—
54C2×C3×C4⋊C4D412yescomplemented
55C2×C3×C22⋊C4—21no—
56#930—21no—
57#930—21no—
58#930—21no—
59#930—21no—
60C2×(C2×D3)⋊C4—22no—
61C2×(C2×D3)⋊C4—24no—
62C2×(C2×D3)⋊C4—21no—
63C2×(C2×D3)⋊C4—21no—
64C2×(C2×D3)⋊C4—21no—
65C2×(C2×D3)⋊C4—21no—
66C2×(C2×D3)⋊C4—21no—
67C2×(C2×D3)⋊C4—21no—
68C2×(C2×D3)⋊C4—21no—
69C2×(C2×D3)⋊C4—22no—
70C2×(C2×D3)⋊C4—44no—
71C2×C4⋊Dic3D411yescharacteristic
72C2×Dic3⋊C4—24no—
73C2×Dic3⋊C4—44no—
74C22×C3⋊C8—21no—
75#882—42no—
76#882—42no—
77C3×C22⋊C8—21no—
78C3×C22⋊C8—21no—
79C3×#95—21no—
80C3⋊#95—22no—
81C3⋊#95—21no—
82C3⋊#95D412yescomplemented
83#830—21no—
84#830D412yescomplemented
85#809—24no—
86#808—24no—
87C2×#121—31no—
88C2×#120—61no—
89C22×C4⋊C4—61no—
90C22×C4⋊C4C2×D311yescharacteristic
91C22×C22⋊C4—31no—
92C22×C22⋊C4—61no—
93#481—62no—
94#481—62no—
95C2×#106—61no—
96C2×#102—61no—
97C2×C22⋊C8—31no—
98C2×C22⋊C8C2×D311yescharacteristic
99#399—62no—
100#399—62no—
101#397—61no—
102#394—62no—
103#392—61no—
104#386—61no—
105#380C2×D311yescharacteristic
106C2×#95—31no—
107#340—22no—
108#327—62no—
109#323—62no—
110C24×C3C2×D411yescharacteristic
111C23×D3—21no—
112C23×D3—21no—
113C23×D3—21no—
114C23×D3—21no—
115C23×D3—21no—
116C23×D3—21no—
117C23×D3—21no—
118C23×D3—42no—
119C23×D3—42no—
120C23×D3SD1612yes—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—42no—
124C22×C3×C4—42no—
125C22×C3×C4—42no—
126C22×C3×C4—41no—
127C22×C3×C4—41no—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4#4111yescharacteristic
134C2×#67—41no—
135C2×#67—41no—
136C2×#67—41no—
137C2×#67—41no—
138C2×#67—41no—
139C2×#67—41no—
140C22×Dic3—24no—
141C22×Dic3—22no—
142C22×Dic3—21no—
143C22×Dic3—21no—
144C22×Dic3—21no—
145C22×Dic3—21no—
146C22×Dic3—42no—
147C22×Dic3—41no—
148C22×Dic3—41no—
149C22×Dic3—42no—
150C2×D12—21no—
151C2×D12—21no—
152C2×D12—42no—
153C2×D12—44no—
154C2×D12—82no—
155C2×D12—82no—
156C2×C3×C8—42no—
157C2×C3×C8—41no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—42no—
161C3×C4⋊C4—41no—
162C3×C4⋊C4—41no—
163C3×C22⋊C4—41no—
164C3×C22⋊C4—41no—
165(C2×D3)⋊C4—48no—
166(C2×D3)⋊C4—44no—
167(C2×D3)⋊C4—41no—
168(C2×D3)⋊C4—41no—
169(C2×D3)⋊C4—41no—
170(C2×D3)⋊C4—41no—
171(C2×D3)⋊C4—41no—
172(C2×D3)⋊C4—41no—
173(C2×D3)⋊C4—82no—
174C4⋊Dic3—21no—
175C4⋊Dic3—21no—
176Dic3⋊C4—48no—
177C2×C3⋊C8—22no—
178C2×C3⋊C8—42no—
179C25—31no—
180C22×D4—31no—
181C22×D4—62no—
182C22×D4—61no—
183C23×C4—21no—
184C23×C4—31no—
185C23×C4—61no—
186C23×C4C22×D311yescharacteristic
187C22×C8—21no—
188C22×C8—61no—
189#121—61no—
190#121—61no—
191#121—61no—
192#121—61no—
193#120—122no—
194#120—122no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—31no—
201C2×C4⋊C4—64no—
202C2×C4⋊C4—124no—
203C2×C4⋊C4D1212yescomplemented
204C2×C4⋊C4#6712yes—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—64no—
207C2×C22⋊C4—62no—
208C2×C22⋊C4—61no—
209C2×C22⋊C4—61no—
210C2×C22⋊C4—61no—
211C2×C22⋊C4—61no—
212C2×C22⋊C4—61no—
213C2×C22⋊C4—61no—
214C2×C22⋊C4—61no—
215C2×C22⋊C4—62no—
216C2×C22⋊C4—124no—
217#106—64no—
218#102—64no—
219C22⋊C8—21no—
220C22⋊C8—21no—
221C22⋊C8—32no—
222C22⋊C8—61no—
223#95—21no—
224#95—32no—
225#95—62no—
226#95—61no—
227C23×C3—22no—
228C23×C3—22no—
229C23×C3#12011yescharacteristic
230C23×C3#12011yescharacteristic
231C23×C3#12111yescharacteristic
232C23×C3C2×SD1611yescharacteristic
233C23×C3#13511yescharacteristic
234C23×C3AGL(1,ℤ/8)11yescharacteristic
235C23×C3#13711yescharacteristic
236C22×D3—22no—
237C22×D3—22no—
238C22×D3—22no—
239C22×D3—44no—
240C22×D3—42no—
241C22×D3—44no—
242C22×D3—42no—
243C22×D3—41no—
244C22×D3—41no—
245C22×D3—41no—
246C22×D3—42no—
247C22×D3—42no—
248C22×D3—41no—
249C22×D3—41no—
250C22×D3—41no—
251C22×D3—41no—
252C22×D3—42no—
253C22×D3—42no—
254C22×D3—82no—
255C22×D3—82no—
256C22×D3—81no—
257C22×D3—81no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—82no—
271C2×C3×C4—82no—
272C2×C3×C4#12011yescharacteristic
273C2×C3×C4#12111yescharacteristic
274C2×C3×C4#12111yescharacteristic
275C2×C3×C4#12711yescharacteristic
276#67—81no—
277#67—81no—
278#67—81no—
279#67—81no—
280C2×Dic3—24no—
281C2×Dic3—44no—
282C2×Dic3—44no—
283C2×Dic3—42no—
284C2×Dic3—42no—
285C2×Dic3—42no—
286C2×Dic3—41no—
287C2×Dic3—41no—
288C2×Dic3—41no—
289C2×Dic3—41no—
290C2×Dic3—42no—
291C2×Dic3—81no—
292D12—42no—
293D12—81no—
294D12—88no—
295C3×C8—81no—
296C3⋊C8—42no—
297C24—32no—
298C24—61no—
299C24—61no—
300C24—61no—
301C24—61no—
302C24—61no—
303C24—61no—
304C24—61no—
305C24—122no—
306C24—122no—
307C24D3×D411yescharacteristic
308C2×D4—61no—
309C2×D4—61no—
310C2×D4—122no—
311C2×D4—124no—
312C2×D4—121no—
313C2×D4—121no—
314C2×D4—121no—
315C2×D4—121no—
316C2×D4—121no—
317C2×D4—121no—
318C2×D4—242no—
319C2×D4—242no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—64no—
328C22×C4—62no—
329C22×C4—61no—
330C22×C4—61no—
331C22×C4—61no—
332C22×C4—61no—
333C22×C4—122no—
334C22×C4—121no—
335C22×C4—121no—
336C22×C4—122no—
337C22×C4C2×D1211yescharacteristic
338C22×C4#23511yescharacteristic
339C22×C4D3×D411yescharacteristic
340C22×C4D3×D411yescharacteristic
341C22×C4D3×D411yescharacteristic
342C22×C4C2×#6711yescharacteristic
343C2×C8—42no—
344C2×C8—41no—
345C2×C8—62no—
346C2×C8—122no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—42no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—61no—
353C4⋊C4—61no—
354C4⋊C4—128no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—128no—
358C22⋊C4—124no—
359C22⋊C4—121no—
360C22⋊C4—121no—
361C22⋊C4—121no—
362C22⋊C4—121no—
363C22⋊C4—121no—
364C22⋊C4—121no—
365C22⋊C4—242no—
366C22×C3—22no—
367C22×C3—22no—
368C22×C3—22no—
369C22×C3—22no—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3#39211yescharacteristic
375C22×C3#44811yescharacteristic
376C22×C3#44911yescharacteristic
377C22×C3#45011yescharacteristic
378C22×C3#45211yescharacteristic
379C22×C3#45312yes—
380C22×C3#45812yes—
381C22×C3#46511yescharacteristic
382C22×C3#46811yescharacteristic
383C2×D3—44no—
384C2×D3—44no—
385C2×D3—42no—
386C2×D3—82no—
387C2×D3—82no—
388C2×D3—82no—
389C2×D3—81no—
390C2×D3—81no—
391C2×D3—82no—
392C2×D3—82no—
393C2×D3—82no—
394C2×D3—81no—
395C2×D3—81no—
396C2×D3—81no—
397C3×C4—21no—
398C3×C4—21no—
399C3×C4—21no—
400C3×C4—21no—
401C3×C4—82no—
402C3×C4—82no—
403Dic3—44no—
404Dic3—81no—
405Dic3—81no—
406C23—22no—
407C23—22no—
408C23—62no—
409C23—62no—
410C23—62no—
411C23—124no—
412C23—122no—
413C23—124no—
414C23—122no—
415C23—121no—
416C23—122no—
417C23—121no—
418C23—122no—
419C23—121no—
420C23—121no—
421C23—121no—
422C23—121no—
423C23—122no—
424C23—121no—
425C23—122no—
426C23—242no—
427C23—242no—
428C23—241no—
429C23—241no—
430C23#88211yescharacteristic
431C23#88411yescharacteristic
432C23D3×SD1611yescharacteristic
433C23#91411yescharacteristic
434C23#91811yescharacteristic
435C23#91911yescharacteristic
436C23#93711yescharacteristic
437D4—122no—
438D4—241no—
439D4—248no—
440D4—241no—
441D4—241no—
442D4—241no—
443D4—241no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—64no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—124no—
460C2×C4—124no—
461C2×C4—122no—
462C2×C4—122no—
463C2×C4—122no—
464C2×C4—121no—
465C2×C4—121no—
466C2×C4—122no—
467C2×C4—121no—
468C2×C4—121no—
469C2×C4—241no—
470C2×C4#88211yescharacteristic
471C2×C4#88411yescharacteristic
472C2×C4#93011yescharacteristic
473C2×C4#94011yescharacteristic
474C8—81no—
475C8—122no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3#161312yes—
481C2×C3#161812yes—
482C2×C3#200011yescharacteristic
483C2×C3#200211yescharacteristic
484C2×C3#202211yescharacteristic
485D3—82no—
486D3—161no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—124no—
496C22—124no—
497C22—122no—
498C22—242no—
499C22—242no—
500C22—242no—
501C22—242no—
502C22—241no—
503C22—241no—
504C22—242no—
505C22—242no—
506C22—241no—
507C22—241no—
508C22—241no—
509C22#480611yescharacteristic
510C22#480711yescharacteristic
511C22#481011yescharacteristic
512C22#481211yescharacteristic
513C22#496011yescharacteristic
514C22#505312yes—
515C22#516911yescharacteristic
516C22#518711yescharacteristic
517C22#519812yes—
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—82no—
523C4—82no—
524C4—124no—
525C4—241no—
526C4—241no—
527C3#992011yescharacteristic, complemented, Sylow
528C2—21no—
529C2—21no—
530C2—21no—
531C2—21no—
532C2—242no—
533C2—481no—
534C2#6953911yescharacteristic
535C2#7027211yescharacteristic
536C2#7032411yescharacteristic
537C2#7116812yes—
538C2#7117712yes—
subgroupG/H
centerC23#937
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1824C2
socleC23×C3#120
prime radical#74031C2
hypercenter#1824D3
hypocenterC3#9920
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#4993C2semidirect
2. C2×#4968C2semidirect
4. C3×#1824C2semidirect
5. C2×#4470C2semidirect
6. C2×#4468C2semidirect
7. #67624C2semidirect
35. #1824D3semidirect
54. C2×C3×C4⋊C4D4semidirect
82. C3⋊#95D4semidirect
84. #830D4semidirect
203. C2×C4⋊C4D12semidirect
527. C3#9920semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#49932
C2×#49682
#740312
C3×#18242
C2×#44702
C2×#44682
#676242
#99203